Choose the incorrect statement

Options

A. Phase constant of SHM depends upon initial conditions
B. Total energy `prop ( "amplitude")^(2)`
C. All motions having same time period are SHM
D. All SHMs have fixed time period

Answer

To solve the question of identifying the incorrect statement regarding simple harmonic motion (SHM), we will analyze each of the provided options step by step.

### Step 1: Analyze the first statement
**Statement:** "Phase constant of SHM depends upon initial condition."

**Analysis:**
The phase constant (φ) in simple harmonic motion does indeed depend on the initial conditions of the system. For example, if the motion starts from the equilibrium position, the phase constant is 0. If it starts from the maximum displacement, the phase constant is π/2. Therefore, this statement is **correct**.

### Step 2: Analyze the second statement
**Statement:** "Total energy is directly proportional to amplitude square."

**Analysis:**
The total energy (E) in SHM is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where \( A \) is the amplitude. This shows that the total energy is indeed directly proportional to the square of the amplitude. Therefore, this statement is **correct**.

### Step 3: Analyze the third statement
**Statement:** "All motions having the same time period are SHM."

**Analysis:**
This statement is **incorrect**. While two different motions can have the same time period, it does not mean both are simple harmonic motions. For example, a simple pendulum and a particle moving in circular motion can have the same time period, but only the simple pendulum exhibits SHM. Therefore, this statement is **incorrect**.

### Step 4: Analyze the fourth statement
**Statement:** "All SHM have a fixed time period."

**Analysis:**
This statement is **correct**. The time period of a simple harmonic oscillator is constant and does not change with time, assuming the system is not damped or driven. Therefore, this statement is **correct**.

### Conclusion
After analyzing all statements, the incorrect statement is:
**"All motions having the same time period are SHM."**